Mathematics

$$\displaystyle \int {{x^3}{e^{{x^2}}}dx = } $$


ANSWER

$$\dfrac{1}{2}\left( {{x^2} - 1} \right){e^{{x^2}}} + c$$


SOLUTION
C)  $$\int x ^ { 3 } e ^ { x ^ { 2 } } d x$$

put $$x ^ { 2 } = t \Rightarrow 2 x d x = d t$$

$$\dfrac { 1 } { 2 } \int t e ^ { t } d t = \dfrac { 1 } { 2 } \left[ \left( t e ^ { t } \right) - \int e ^ { t } d t \right]$$                {integration by parts}

$$= \dfrac { 1 } { 2 } \left[ t e ^ { t } - e ^ { t } \right]+c$$

$$= \dfrac { 1 } { 2 } \left( x ^ { 2 } e ^ { x ^ { 2 } } - e ^ { x ^ { 2 } } \right) + c$$

$$= \dfrac { 1 } { 2 } \left( x ^ { 2 } - 1 \right) e ^ { x ^ { 2 } } + c$$
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Single Correct Medium Published on 17th 09, 2020
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